The geometric dimension of some small configurations

Abstract

Recently, Jungnickel and Tonchev (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012) introduced new invariants for simple incidence structures \({\mathcal{D}}\), which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of \({\mathcal{D}}\) into projective geometries \({\Pi} = PG(n, q)\), where an embedding means identifying the points of \({\mathcal{D}}\) with a point set V in \({\Pi}\) in such a way that every block of \({\mathcal{D}}\) is induced as the intersection of V with a suitable subspace of \({\Pi}\). Then the new invariant, the geometric dimension\({\mathrm{geomdim}_{q}\mathcal{D}}\) of \({\mathcal{D}}\), is the smallest value of n for which \({\mathcal{D}}\) may be embedded into the n-dimensional projective geometry PG(n, q). It is the aim of this paper to discuss a few additional general results regarding these invariants, and to determine them for some further examples, mainly some small configurations; this will answer some problems posed in (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012).